What is the general solution for tan^2(x)+4tan(x)+3=0 ?

The general solution for tan^2(x)+4tan(x)+3=0 is x=(3pi)/4+pin,x=-1.24904…+pin

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Popular >

tan^2(x)+4tan(x)+3=0

Solution

tan^{2} (x) + 4 tan (x) + 3 = 0

Solution

x = \frac{3 π}{4} + πn, x = - 1.24904 \dots + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = - 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + 4 tan (x) + 3 = 0

Solve by substitution

tan^{2} (x) + 4 tan (x) + 3 = 0

Let:

tan (x) = u

u^{2} + 4 u + 3 = 0

u^{2} + 4 u + 3 = 0 : u = - 1, u = - 3

u^{2} + 4 u + 3 = 0

Solve with the quadratic formula

u^{2} + 4 u + 3 = 0

Quadratic Equation Formula:

For

a = 1, b = 4, c = 3

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

4^{2} - 4 \cdot 1 \cdot 3 = 2

4^{2} - 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 4^{2} - 12

4^{2} = 16

= 16 - 12

Subtract the numbers:

16 - 12 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- 4 \pm 2}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 4 + 2}{2 \cdot 1}, u_{2} = \frac{- 4 - 2}{2 \cdot 1}

u = \frac{- 4 + 2}{2 \cdot 1} : - 1

\frac{- 4 + 2}{2 \cdot 1}

Add/Subtract the numbers:

- 4 + 2 = - 2

= \frac{- 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- 4 - 2}{2 \cdot 1} : - 3

\frac{- 4 - 2}{2 \cdot 1}

Subtract the numbers:

- 4 - 2 = - 6

= \frac{- 6}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

The solutions to the quadratic equation are:

u = - 1, u = - 3

Substitute back

u = tan (x)

tan (x) = - 1, tan (x) = - 3

tan (x) = - 1, tan (x) = - 3

tan (x) = - 1 : x = \frac{3 π}{4} + πn

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

tan (x) = - 3 : x = arctan (- 3) + πn

tan (x) = - 3

Apply trig inverse properties

tan (x) = - 3

General solutions for

tan (x) = - 3

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 3) + πn

x = arctan (- 3) + πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = arctan (- 3) + πn

Show solutions in decimal form

x = \frac{3 π}{4} + πn, x = - 1.24904 \dots + πn

Popular Examples

(1+tan(x))^2+(1-tan(x))^2=sec^2(x)

(1 + tan (x))^{2} + (1 - tan (x))^{2} = sec^{2} (x)

tan(x)=-(sqrt(2))/2

tan (x) = - \frac{2}{2}

cot(x)=1.2

cot (x) = 1.2

cos(3/2 pi-x)+sin(4x)=0

cos (\frac{3}{2} π - x) + sin (4 x) = 0

tan(x)= 30/100

tan (x) = \frac{30}{100}

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+4tan(x)+3=0 ?
The general solution for tan^2(x)+4tan(x)+3=0 is x=(3pi)/4+pin,x=-1.24904…+pin